- Research Article
- Open access
- Published:
A Note on Stability of a Linear Functional Equation of Second Order Connected with the Fibonacci Numbers and Lucas Sequences
Journal of Inequalities and Applications volume 2010, Article number: 793947 (2010)
Abstract
We prove the Hyers-Ulam stability of a second-order linear functional equation in single variable (with constant coefficients) that is connected with the Fibonacci numbers and Lucas sequences. In this way we complement, extend, and/or improve some recently published results on stability of that equation.
1. Introduction
In this paper , , , and stand, as usual, for the sets of complex numbers, real numbers, integers, and positive integers, respectively. Let be a nonempty set, , be a Banach space over a field , , , and denote the complex roots of the equation
Moreover, , , and (only for bijective ) for and .
The problem of stability of functional equations was motivated by a question of Ulam asked in 1940 and a solution to it by Hyers published in [1]. Since then numerous papers have been published on that subject and we refer to [2–7] for more details, some discussions and further references; for examples of very recent results see, for example, [8–12]. Jung has proved in [5] (see also [13]) some results on solutions and stability of the functional equation
in the case where and for . The result on stability (see [5, Theorem ]) can be stated as follows.
Theorem 1.1.
Let , , , , , and satisfy the inequality
Then there is a unique solution of the functional equation
with
If and , then solutions of the difference equation (1.4) are called the Lucas sequences (see, e.g., [14]); in some special cases they are called with specific names; for example; the Fibonacci numbers (, , and ), the Lucas numbers (, , and ), the Pell numbers (, , and ), the Pell-Lucas (or companion Lucas) numbers (, , and ), and the Jacobsthal numbers (, , and ).
For some information and further references concerning the functional equations in single variable we refer to [15–17]; for an ample survey on stability results for those equations see [2]. Let us mention yet that the problem of stability of functional equations is connected to the notions of controlled chaos (see [18]) and shadowing (see [19–21]).
Remark 1.2.
If is bijective, then, with , (1.2) can be written in the following equivalent form:
Clearly, (1.1) is the characteristic equation of (1.6).
In view of Remark 1.2, from [22,Theorem ] (see also [23]) the following stability result, concerning (1.2), can be derived.
Theorem 1.3.
Let for , be bijective, , and satisfy the inequality
Then there is a unique solution of (1.2) with
Theorem 1.3 appears to be much more general than Theorem 1.1 (obtained by a different method of proof). But on the other hand, estimation (1.5) is significantly sharper than (1.8) in numerous cases (take, e.g., and , with some large ). Therefore, there arises a natural question if the method applied in [5] can be modified so as to prove a more general equivalent of Theorem 1.3, but with an estimation better than (1.8). In this paper, we show that this is the case. Namely, we prove the following.
Theorem 1.4.
Let and satisfy inequality (1.7). Suppose that and one of the following two conditions is valid:
() for ;
() for and is bijective.
Then there exists a solution of (1.2) such that
Moreover, if condition is valid, then there exists exactly one solution of (1.2) with .
Remark 1.5.
Note that, for bijective , Theorem 1.4 improves estimation (1.8) in some cases (take, e.g., , , or , ); however, in some other situations (e.g., , ), estimation (1.8) is better. Theorem 1.4 also complements Theorem 1.3 because can be quite arbitrary in the case of .
2. Proof of Theorem 1.4
Clearly, and . We start with the case .
Fix and first assume that . Write
Then, for each and ,
whence
and consequently
This means that, for each , is a Cauchy sequence and therefore there exists the limit . Further, for every ,
and, by (2.4) with and ,
Now, assume that . This means that is bijective. Let
Then, for each and ,
and next, by (1.7),
Hence,
So, for each , is a Cauchy sequence and consequently there exists the limit . Note that, for every , (2.5) holds and, by (2.10) with and ,
Thus, we have proved that, for , inequality (2.6) holds and is a solution to (1.2). Define by
Then, for , it follows from (2.5) that
and, by (1.1) and (2.6),
In the case where is bijective, the uniqueness of results from [22, Proposition ], in view of Remark 1.2.
Now, assume that . Then (see, e.g., [24, page 39], [25], or [26, 27]) is a complex Banach space with the linear structure and the Taylor norm given by
Clearly, for all .
Define by for . Then,
So, by the previous part of the proof, there exists a solution of (1.2) such that
Write for , . Clearly, , given by for , is a solution of (1.2), and (1.9) holds.
It remains to prove the statement concerning uniqueness of . So, let be a solution of (1.2) with . Let for . It is easily seen that is a solution of (1.2). Moreover, for every ,
Hence, by [22, Proposition ], , which yields .
3. Consequences of Theorem 1.4
Now we present some consequences of Theorem 1.4 and some results from [22, 28, 29].
Theorem 3.1.
Let and satisfy (1.7). Suppose that one of the following three conditions is valid:
(i) for and ;
(ii) for and is bijective;
(iii)(ii) holds and .
Then there exists a solution of (1.2) such that
where
Moreover, if for , then there exists exactly one solution of (1.2) such that .
Proof.
If (i) is valid, then Theorem 1.4 yields (3.1) with . Further, by (1.7),
and for are roots of the equation
Hence, by [22, Theorem ], there is a solution of the functional equation
such that
(The last equality is due to the fact that .) It is easily seen that is a solution to (1.2).
Next, consider the case of (ii). Then, in view of Theorem 1.4, there is a solution of (1.2) satisfying (3.1) with . Further,
with . Hence, according to [22, Theorem ], there exists a function satisfying (1.6) and inequality (3.1), with . Now, it is enough to note that is a solution to (1.2), as well.
Finally, if (iii) holds, then it is enough to use [22, Theorem ] and Theorem 1.4 (the case of ).
The statement concerning uniqueness results from [22, Proposition ].
Remark 3.2.
If for some (or, equivalently, for some ), then (1.2) can be nonstable, by which we mean that there is a function such that (1.7) holds with some real and for each solution of (1.2) (see, e.g., [28], [22, Example ], or [29]).
Remark 3.3.
Note that, in the case where are real numbers, we have
with
4. Some Critique and Final Remarks
Functional equation (1.2) has been patterned on difference equation (1.4). However, if we want to apply the results of Theorems 1.1–3.1 to the Lucas sequences we come across two obstacles. The first one concerns the domain of and arises from the difference equation (1.4) being written in "wrong" historical form, inconsistent with the general concept of functional equations. Actually it should be written as the functional equation
which corresponds to (1.6). The second obstacle is connected with the restrictions on . For some interesting cases (Fibonacci, Lucas, or Pell numbers), we have (or, if somebody prefers, ) and such case is not covered if is not bijective (which is the case when and or, equivalently, ). All these obstacles can be overcome if, instead of Theorems 1.1–3.1, we use the following result derived from [29, Theorem ].
Proposition 4.1.
Let , , and for , , and
Then there is a solution of (4.1) that satisfies (1.8).
For instance, if and (the case of the Fibonacci and Lucas numbers), we have the following.
Corollary 4.2.
Let , , , and be a sequence in with
Then there is a sequence in such that
Proof.
Note that . Thus, by Proposition 4.1, there is a sequence in such that (4.4) is valid.
Remark 4.3.
If and (the case of Jacobsthal numbers), then one of the roots of (1.1) is equal to and therefore (4.1) is not stable (see [28]), by which we mean that, for each , there is such that and for every solution of (1.6); for such function can be chosen with, for example, and (in [28, the proofs of Lemma and Theorem ] take ).
References
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
Agarwal RP, Xu B, Zhang W: Stability of functional equations in single variable. Journal of Mathematical Analysis and Applications 2003, 288(2):852–869. 10.1016/j.jmaa.2003.09.032
Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.
Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.
Jung S-M: Functional equation and its Hyers-Ulam stability. Journal of Inequalities and Applications 2009, 2009:-10.
Moszner Z: On the stability of functional equations. Aequationes Mathematicae 2009, 77(1–2):33–88. 10.1007/s00010-008-2945-7
Paneah B: A new approach to the stability of linear functional operators. Aequationes Mathematicae 2009, 78(1–2):45–61. 10.1007/s00010-009-2956-z
Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.
Chudziak J: Stability problem for the Gołąb-Schinzel type functional equations. Journal of Mathematical Analysis and Applications 2008, 339(1):454–460. 10.1016/j.jmaa.2007.07.006
Ciepliński K: Stability of the multi-Jensen equation. Journal of Mathematical Analysis and Applications 2010, 363(1):249–254. 10.1016/j.jmaa.2009.08.021
Miheţ D: The probabilistic stability for a functional equation in a single variable. Acta Mathematica Hungarica 2009, 123(3):249–256. 10.1007/s10474-008-8101-y
Sikorska J: On a pexiderized conditional exponential functional equation. Acta Mathematica Hungarica 2009, 125(3):287–299. 10.1007/s10474-009-9019-8
Jung S-M: Hyers-Ulam stability of Fibonacci functional equation. Bulletin of the Iranian Mathematical Society 2009, 35(2):217–227, 282.
Ribenboim P: My Numbers, My Friends, Popular Lectures on Number Theory. Springer, New York, NY, USA; 2000:xii+375.
Baron K, Jarczyk W: Recent results on functional equations in a single variable, perspectives and open problems. Aequationes Mathematicae 2001, 61(1–2):1–48. 10.1007/s000100050159
Kuczma M: Functional Equations in a Single Variable, Monografie Matematyczne. Volume 46. Polish Scientific Publishers, Warszawa, Poland; 1968:383 pp.
Kuczma M, Choczewski B, Ger R: Iterative Functional Equations, Encyclopedia of Mathematics and Its Applications. Volume 32. Cambridge University Press, Cambridge, UK; 1990:xx+552.
Stević S: Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos. Chaos, Solitons & Fractals 2008, 35(2):238–245. 10.1016/j.chaos.2007.07.037
Hayes W, Jackson KR: A survey of shadowing methods for numerical solutions of ordinary differential equations. Applied Numerical Mathematics 2005, 53(2–4):299–321.
Palmer K: Shadowing in Dynamical Systems, Mathematics and Its Applications. Volume 501. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xiv+299.
Pilyugin SY: Shadowing in Dynamical Systems, Lecture Notes in Mathematics. Volume 1706. Springer, Berlin, Germany; 1999:xviii+271.
Brzdęk J, Popa D, Xu B: Hyers-Ulam stability for linear equations of higher orders. Acta Mathematica Hungarica 2008, 120(1–2):1–8. 10.1007/s10474-007-7069-3
Trif T: Hyers-Ulam-Rassias stability of a linear functional equation with constant coefficients. Nonlinear Functional Analysis and Applications 2006, 11(5):881–889.
Fabian M, Habala P, Hájek P, Montesinos Santalucía V, Pelant J, Zizler V: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer, New York, NY, USA; 2001:x+451.
Ferrera J, Muñoz GA: A characterization of real Hilbert spaces using the Bochnak complexification norm. Archiv der Mathematik 2003, 80(4):384–392.
Kadison RV, Ringrose JR: Fundamentals of the Theory of Operator Algebras. Vol. I. Elementary Theory, Pure and Applied Mathematics. Volume 100. Academic Press, New York, NY, USA; 1983:xv+398.
Kadison RV, Ringrose JR: Fundamentals of the Theory of Operator Algebras. Vol. I. Elementary Theory, Graduate Studies in Mathematics. Volume 15. American Mathematical Society, Providence, RI, USA; 1997:xvi+398.
Brzdęk J, Popa D, Xu B: Note on nonstability of the linear recurrence. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2006, 76: 183–189. 10.1007/BF02960864
Popa D: Hyers-Ulam stability of the linear recurrence with constant coefficients. Advances in Difference Equations 2005, 2005(2):101–107. 10.1155/ADE.2005.101
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science, and Technology (no. 2010-0007143).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Brzdęk, J., Jung, SM. A Note on Stability of a Linear Functional Equation of Second Order Connected with the Fibonacci Numbers and Lucas Sequences. J Inequal Appl 2010, 793947 (2010). https://doi.org/10.1155/2010/793947
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/793947